t-tests for population mean µ (when σ is unknown)
1. State Hypotheses:
H 0 : µ = µ0
Ha : µ > µ0
Ha : µ < µ0
Ha : µ 6= µ0
2. Test Statistic:
t=
x̄ − µ0
s
√
n
3. p-value: probability of getting more extreme test statistic than
t given H0 is true
Under the assumption that H0 is true, the test statistic follows a tdistribution with df = n − 1
Ha : µ > µ 0
p-value = P (T > t)
Ha : µ < µ 0
p-value = P (T < t)
=
P (T > −t)
|
{z
}
right area, easier f or t
Ha : µ 6= µ0
p-value = 2P (T > |t|)
4. Decision: Reject H0 if
p-value ≤ α
5. Conclusion: In terms of the problem and Ha
If reject H0 ⇒ statistically significant evidence to support the claim
made in Ha
If fail to reject H0 ⇒ no statistically significant evidence to support the
claim made in Ha
1
Example 1: A random sample of 30 pills yielded a mean level of 20.5
mg of aspirin and a standard deviation of 1.5 mg. Assuming the pill is
supposed to contain 20 mg of aspirin, are these sample pills acceptable
at the 0.02 level?
1. State the null and alternative hypotheses:
2. Compute the value of the test statistic:
3. Find the p-value associated with the test statistic:
4. Decision:
5. There is
evidence (at the α = 0.01 level) to conclude that the pills have an average
aspirin amount that differs from 20 mg.
2
Example 2: A pain reliever currently being used in a hospital is known
to bring relief to patients with a mean time of 3.5 minutes. A new drug
administered to a random sample of 50 patients, yields a mean time
to relief of 2.8 minutes and a the standard deviation of 1.14 minutes.
Is there sufficient evidence that the new drug lowers the mean time to
pain relief? Test at the α = 0.05 level.
1. State the null and alternative hypotheses:
2. Compute the value of the test statistic:
3. Find the p-value associated with the test statistic:
4. Decision:
ev5. There is
idence (at the α = 0.05 level) to conclude that the new drug lowers the
mean time to pain relief.
3